Determine logaritmic function based on samples of unique visitors I need to estimate the unique visitors for any given number of days. I have the unique visitors per day, day, week and month.
 1 day :  20000 visitors
 7 days:  85000 visitors
30 days: 200000 visitors

How can I determine a logaritmic function that has these 3 points?
 A: Logarithmic functions have trouble with $0$, so you will have to exclude that one.  I just fed the other three to Excel and found visitors$=51856 \ln t+ 9240$ but it is not a very good fit.  Much better is an power law:  visitors$=20748t^{0.6806}$
A: We should expect the unique visitors to grow quickly the first days.
The visitors coming back after a long period is relatively small, but there will always be a group of new visitors. So in the long run the numbers should be almost linear.
This would suggest a logarithmic function like:
$$
a*log(x+1) + bx
$$

Determining a and b this function based on samples is fairly hard. However the function can be approached for $x > 1$ with:
$$
\sqrt{ax^2+bx}
$$

a and b can be calculated using Quadatic regression.
With 3 samples and $y=0$ on $x=0$ the function would not exactly pass through all 3 points, but it'd be a best fit. Taken that we don't really care about $x<1$, the best is to introduce c having the error biggest on $t=0$ and becoming less relevant for a bigger x.
$$
\sqrt{ax^2+bx+c}
$$
We calculate a, b and c using the discriminant.
$$
a = (x3*(y2-y1) + x2*(y1-y3) + x1*(y3-y2)) / ((x1-x2) * (x1-x3) * (x2-x3))
$$
$$
b = (x3^2(y1-y2) + x1^2(y2-y3) + x2^2*(y3-y1) )/ ((x1-x2) * (x1-x3) * (x2-x3))
$$
$$
c = (x3*(x2*(x2-x3)*y1 + x1*(x3-x1)*y2) + x1*(x1-x2)*x2*y3) / ((x1-x2) * (x1-x3) * (x2-x3))
$$
This answer is courtesy of Nico Schoonderwoerd aka @klup
